Finite Group Databasebeta

← #31792608

Subgroups — #31792608

Order 17536.

462 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22578111C211yesmaximal, characteristic, complemented
2#22578037C211yesmaximal, characteristic
3#22577748C211yesmaximal, characteristic, complemented
4#22577723C211yesmaximal, characteristic, complemented
5#22577719C211yesmaximal, characteristic, complemented
6#22577692C211yesmaximal, characteristic, complemented
7#22577635C211yesmaximal, characteristic, complemented
8#22577601C211yesmaximal, characteristic, complemented
9#22577553C211yesmaximal, characteristic, complemented
10#22577552C211yesmaximal, characteristic, complemented
11#22577488C211yesmaximal, characteristic, complemented
12C8×C22⋊D137C211yesmaximal, characteristic, complemented
13C2×#17984010C211yesmaximal, characteristic, complemented
14#102×D137C211yesmaximal, characteristic, complemented
15#465×C137C211yesmaximal, characteristic, complemented
16C22×D548C2211yescharacteristic
17C2×D4×D137—21no—
18C2×D4×D137C2211yescharacteristic
19C2×#14555497C2211yescharacteristic
20C2×#14555497C2211yescharacteristic, complemented
21C2×C8×D137C2211yescharacteristic, complemented
22#17984087C2211yescharacteristic, complemented
23#17984084C2211yescharacteristic
24#17984078—21no—
25#17984068—21no—
26#17984065—21no—
27#17984064C2211yescharacteristic, complemented
28#17984063C2211yescharacteristic
29#17984061—21no—
30#17984058—21no—
31#17984056—21no—
32Q8×Dic137C2211yescharacteristic, complemented
33#17984050C2211yescharacteristic
34C4×C22⋊D137C2211yescharacteristic
35#17984048—21no—
36C2×C4⋊Dic137C2211yescharacteristic, complemented
37C4⋊C4×D137C2211yescharacteristic, complemented
38#17984019C2211yescharacteristic, complemented
39#17984019C2211yescharacteristic, complemented
40#17984017C2211yescharacteristic, complemented
41#17984010C2211yescharacteristic
42#17984010C2211yescharacteristic
43#17984010C2211yescharacteristic, complemented
44#17984010C2211yescharacteristic, complemented
45#17984009C2211yescharacteristic, complemented
46#17984009C2211yescharacteristic, complemented
47#17984008C2211yescharacteristic
48#17984008C2211yescharacteristic, complemented
49C22⋊#11764040C2211yescharacteristic, complemented
50#17984000C2211yescharacteristic, complemented
51C8×Dic137C2211yescharacteristic, complemented
52C2×SD16×C137C2211yescharacteristic, complemented
53C22×C8×C137C2211yescharacteristic, complemented
54C22⋊Q8×C137C2211yescharacteristic, complemented
55#121×C137C2211yescharacteristic
56#106×C137C2211yescharacteristic, complemented
57#103×C137C2211yescharacteristic, complemented
58#102×C137C2211yescharacteristic, complemented
59C23×D137—21no—
60C23×D137—21no—
61C2×Dic274—21no—
62C2×D548—22no—
63C2×D548—21no—
64C2×D548—21no—
65C2×D548—41no—
66C2×D548C2311yescharacteristic
67C2×D548C2311yescharacteristic
68C2×C22⋊D137—21no—
69C2×C22⋊D137—21no—
70C2×C22⋊D137—21no—
71C2×C22⋊D137D411yescharacteristic
72D4×D137—22no—
73D4×D137—41no—
74D4×D137—44no—
75C22×Dic137—21no—
76C2×C4×D137—21no—
77C2×C4×D137—21no—
78C2×C4×D137C2311yescharacteristic
79#14555497—22no—
80#14555497—41no—
81C2×#11764040C2311yescharacteristic
82C2×#11764040C2311yescharacteristic
83C8×D137—41no—
84C4⋊Dic137—21no—
85C4⋊Dic137—21no—
86C4⋊Dic137C2311yescharacteristic
87C4⋊Dic137C2311yescharacteristic
88#14555489—21no—
89#14555489—21no—
90#14555489—21no—
91#14555489D411yescharacteristic
92C22⋊Dic137D411yescharacteristic
93#14555487—21no—
94#14555487—21no—
95#14555487—21no—
96#14555487D411yescharacteristic
97C4×Dic137—21no—
98C4×Dic137C2311yescharacteristic
99C2×Q8×C137C2311yescharacteristic
100C2×D4×C137—21no—
101C2×D4×C137C2311yescharacteristic
102C22×C4×C137C2311yescharacteristic
103SD16×C137—41no—
104C2×C8×C137—41no—
105C2×C8×C137C2311yescharacteristic
106C2×C8×C137C2311yescharacteristic
107C4⋊C4×C137—21no—
108C4⋊C4×C137C2311yescharacteristic
109C4⋊C4×C137C2311yescharacteristic
110C22⋊C4×C137—21no—
111C22⋊C4×C137—21no—
112C22×D137—21no—
113C22×D137—21no—
114C22×D137—42no—
115C22×D137—42no—
116C22×D137—42no—
117C22×D137—42no—
118C22×D137—81no—
119C22×D137—81no—
120C22×D137C2×D411yescharacteristic
121Dic274—81no—
122D548—21no—
123D548—21no—
124D548—21no—
125D548—42no—
126D548—42no—
127D548#4112yes—
128C22⋊D137—42no—
129C22⋊D137—42no—
130C22⋊D137—81no—
131C22⋊D137SD1614yescomplemented
132C2×Dic137—21no—
133C2×Dic137—21no—
134C2×Dic137—81no—
135C2×Dic137C2×D411yescharacteristic
136C2×Dic137C2×D411yescharacteristic
137C2×Dic137C2×D411yescharacteristic
138C4×D137—21no—
139C4×D137—21no—
140C4×D137—42no—
141C4×D137—81no—
142#11764040—21no—
143#11764040—21no—
144C23×C137—21no—
145C23×C137C2×D411yescharacteristic
146Q8×C137—21no—
147Q8×C137—21no—
148D4×C137—21no—
149D4×C137—21no—
150D4×C137—42no—
151C2×C4×C137—21no—
152C2×C4×C137—21no—
153C2×C4×C137—21no—
154C2×C4×C137—21no—
155C2×C4×C137—21no—
156C2×C4×C137C2×D411yescharacteristic
157C2×C4×C137C2411yescharacteristic
158C8×C137—21no—
159C8×C137—21no—
160C22×C137—21no—
161C22×C137—21no—
162C22×C137—41no—
163C22×C137—41no—
164C22×C137—41no—
165C22×C137C2×SD1612yes—
166C22×C137C22×D411yescharacteristic
167C2×D137—21no—
168C2×D137—21no—
169C2×D137—22no—
170C2×D137—41no—
171C2×D137—41no—
172C2×D137—41no—
173C2×D137—41no—
174C2×D137—41no—
175C2×D137—84no—
176C2×D137—84no—
177C2×D137C2×SD1612yes—
178C4×C137—21no—
179C4×C137—41no—
180C4×C137—41no—
181C4×C137—41no—
182C4×C137C2×#4111yescharacteristic
183C4×C137H(5,𝔽2)11yescharacteristic
184Dic137—21no—
185Dic137—21no—
186Dic137—41no—
187Dic137—41no—
188Dic137C2×SD1612yes—
189C2×C137—22no—
190C2×C137—81no—
191C2×C137#54611yescharacteristic
192C2×C137C22×SD1611yescharacteristic
193C2×C137#57611yescharacteristic
194D137—22no—
195D137—42no—
196D137—81no—
197C137#329511yescharacteristic, complemented, Sylow
198#3295—1371nomaximal, Sylow
199#554—1371no—
200D42—1371no—
201#486—1371no—
202#481—1371no—
203#481—1371no—
204#476—1371no—
205#465—1371no—
206#465D13711yescharacteristic, complemented
207#460—1371no—
208#450—1371no—
209#450—1371no—
210C4×SD16—1371no—
211C8×D4—1371no—
212C2×#102—1371no—
213C2×#102—1371no—
214C22×D4—2741no—
215C22×D4—1371no—
216C22×D4—1371no—
217C2×SD16—1371no—
218C2×SD16—1371no—
219C2×SD16C2×D13711yescharacteristic, complemented
220C22×C8—1371no—
221C22×C8C2×D13711yescharacteristic, complemented
222C4⋊Q8—1371no—
223#127—1371no—
224#125—2741no—
225C22⋊Q8—2741no—
226C22⋊Q8—2741no—
227C22⋊Q8—1371no—
228C22⋊Q8C2×D13711yescharacteristic, complemented
229#121—2741no—
230#121—1371no—
231#121C2×D13711yescharacteristic
232#120—2741no—
233#120—2741no—
234C4×Q8—1371no—
235C4×D4—2741no—
236C4×D4—1371no—
237C4×D4—1371no—
238C2×C4⋊C4—1371no—
239C2×C4⋊C4—1371no—
240#106—1371no—
241#106—1371no—
242#106C2×D13711yescharacteristic, complemented
243C4⋊C8—1371no—
244#103C2×D13711yescharacteristic, complemented
245#102—1371no—
246#102—1371no—
247#102—1371no—
248#102—1371no—
249#102—1371no—
250#102—1371no—
251#102—1371no—
252#102—1371no—
253#102C2×D13711yescharacteristic, complemented
254C22⋊C8—1371no—
255C22⋊C8—1371no—
256C4×C8—1371no—
257C24—2741no—
258C24—2741no—
259C2×Q8—2741no—
260C2×Q8C22×D13711yescharacteristic
261C2×D4—21no—
262C2×D4—2742no—
263C2×D4—2741no—
264C2×D4—2742no—
265C2×D4—2741no—
266C2×D4—2741no—
267C2×D4—2741no—
268C2×D4—2741no—
269C2×D4—5481no—
270C2×D4—5481no—
271C2×D4—5484no—
272C2×D4—1371no—
273C2×D4—1371no—
274C2×D4—1371no—
275C2×D4C22×D13711yescharacteristic
276C22×C4—2741no—
277C22×C4—2741no—
278C22×C4—2741no—
279C22×C4—1371no—
280C22×C4C22×D13711yescharacteristic
281SD16—41no—
282SD16—2742no—
283SD16—5481no—
284C2×C8—41no—
285C2×C8—5481no—
286C2×C8—1371no—
287C2×C8—1371no—
288C2×C8C22×D13711yescharacteristic
289C2×C8C22×D13711yescharacteristic
290C4⋊C4—21no—
291C4⋊C4—2741no—
292C4⋊C4—2741no—
293C4⋊C4—2741no—
294C4⋊C4—2741no—
295C4⋊C4—2741no—
296C4⋊C4—1371no—
297C4⋊C4—1371no—
298C4⋊C4—1371no—
299C4⋊C4C22×D13711yescharacteristic
300C4⋊C4C22×D13711yescharacteristic
301C22⋊C4—21no—
302C22⋊C4—21no—
303C22⋊C4—2741no—
304C22⋊C4—2741no—
305C22⋊C4—2741no—
306C22⋊C4—1371no—
307C22⋊C4—1371no—
308C42—2741no—
309C42—1371no—
310C23—21no—
311C23—2741no—
312C23—2741no—
313C23—5482no—
314C23—5482no—
315C23—5482no—
316C23—5482no—
317C23—10961no—
318C23—10961no—
319C23—1371no—
320C23D4×D13711yescharacteristic
321Q8—21no—
322Q8—21no—
323Q8—10961no—
324D4—21no—
325D4—21no—
326D4—42no—
327D4—2741no—
328D4—2741no—
329D4—2741no—
330D4—5482no—
331D4—5482no—
332D4—5482no—
333D4—5482no—
334D4—10961no—
335D4—1374no—
336D4—1372no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—2741no—
343C2×C4—2741no—
344C2×C4—2741no—
345C2×C4—2741no—
346C2×C4—5482no—
347C2×C4—10961no—
348C2×C4—10961no—
349C2×C4—1371no—
350C2×C4—1371no—
351C2×C4—1371no—
352C2×C4D4×D13711yescharacteristic
353C2×C4C23×D13711yescharacteristic
354C8—21no—
355C8—21no—
356C8—2741no—
357C8—2741no—
358C22—21no—
359C22—21no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—2741no—
364C22—2741no—
365C22—2742no—
366C22—5481no—
367C22—5481no—
368C22—5481no—
369C22—5481no—
370C22—5481no—
371C22—10964no—
372C22—10964no—
373C22—1372no—
374C22SD16×D13712yes—
375C22C2×D4×D13711yescharacteristic
376C4—21no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—2741no—
381C4—2741no—
382C4—5481no—
383C4—5481no—
384C4—1372no—
385C4#41×D13711yescharacteristic
386C4#1798413311yescharacteristic
387C2—22no—
388C2—81no—
389C2—2742no—
390C2—5482no—
391C2—10961no—
392C2#2257804211yescharacteristic
393C2C2×SD16×D13711yescharacteristic
394C2#2257820511yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#465×C137C2
socleC22×C137C22×D4
prime radical#22578037C2
hypercenter#465D137
hypocenterC137#3295
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #22578111C2semidirect
3. #22577748C2semidirect
4. #22577723C2semidirect
5. #22577719C2semidirect
6. #22577692C2semidirect
7. #22577635C2semidirect
8. #22577601C2semidirect
9. #22577553C2semidirect
10. #22577552C2semidirect
11. #22577488C2semidirect
12. C8×C22⋊D137C2semidirect
13. C2×#17984010C2semidirect
14. #102×D137C2semidirect
15. #465×C137C2semidirect
20. C2×#14555497C22semidirect
21. C2×C8×D137C22semidirect
22. #17984087C22semidirect
27. #17984064C22semidirect
32. Q8×Dic137C22semidirect
36. C2×C4⋊Dic137C22semidirect
37. C4⋊C4×D137C22semidirect
38. #17984019C22semidirect
39. #17984019C22semidirect
40. #17984017C22semidirect
43. #17984010C22semidirect
44. #17984010C22semidirect
45. #17984009C22semidirect
46. #17984009C22semidirect
48. #17984008C22semidirect
49. C22⋊#11764040C22semidirect
50. #17984000C22semidirect
51. C8×Dic137C22semidirect
52. C2×SD16×C137C22semidirect
53. C22×C8×C137C22semidirect
54. C22⋊Q8×C137C22semidirect
56. #106×C137C22semidirect
57. #103×C137C22semidirect
58. #102×C137C22semidirect
131. C22⋊D137SD16semidirect
197. C137#3295semidirect
206. #465D137semidirect
219. C2×SD16C2×D137semidirect
221. C22×C8C2×D137semidirect
228. C22⋊Q8C2×D137semidirect
242. #106C2×D137semidirect
244. #103C2×D137semidirect
253. #102C2×D137semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#225781112
#225780372
#225777482
#225777232
#225777192
#225776922
#225776352
#225776012
#225775532
#225775522
#225774882
C8×C22⋊D1372
C2×#179840102
#102×D1372
#465×C1372
#3295137